LMFDB - Embedded newform 1450.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(1,1450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.5783082931$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	1450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.81361 q^{3} +1.00000 q^{4} +1.81361 q^{6} +2.52444 q^{7} +1.00000 q^{8} +0.289169 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.81361 q^{3} +1.00000 q^{4} +1.81361 q^{6} +2.52444 q^{7} +1.00000 q^{8} +0.289169 q^{9} -2.91638 q^{11} +1.81361 q^{12} +3.10278 q^{13} +2.52444 q^{14} +1.00000 q^{16} +6.91638 q^{17} +0.289169 q^{18} +1.81361 q^{19} +4.57834 q^{21} -2.91638 q^{22} -5.68111 q^{23} +1.81361 q^{24} +3.10278 q^{26} -4.91638 q^{27} +2.52444 q^{28} +1.00000 q^{29} -8.12193 q^{31} +1.00000 q^{32} -5.28917 q^{33} +6.91638 q^{34} +0.289169 q^{36} +7.39194 q^{37} +1.81361 q^{38} +5.62721 q^{39} +8.39194 q^{41} +4.57834 q^{42} -5.04888 q^{43} -2.91638 q^{44} -5.68111 q^{46} +4.86751 q^{47} +1.81361 q^{48} -0.627213 q^{49} +12.5436 q^{51} +3.10278 q^{52} +5.30833 q^{53} -4.91638 q^{54} +2.52444 q^{56} +3.28917 q^{57} +1.00000 q^{58} +2.60806 q^{59} -10.0680 q^{61} -8.12193 q^{62} +0.729988 q^{63} +1.00000 q^{64} -5.28917 q^{66} -2.68111 q^{67} +6.91638 q^{68} -10.3033 q^{69} +5.68111 q^{71} +0.289169 q^{72} -15.0680 q^{73} +7.39194 q^{74} +1.81361 q^{76} -7.36222 q^{77} +5.62721 q^{78} -1.73501 q^{79} -9.78389 q^{81} +8.39194 q^{82} -13.1219 q^{83} +4.57834 q^{84} -5.04888 q^{86} +1.81361 q^{87} -2.91638 q^{88} +5.23527 q^{89} +7.83276 q^{91} -5.68111 q^{92} -14.7300 q^{93} +4.86751 q^{94} +1.81361 q^{96} +1.27001 q^{97} -0.627213 q^{98} -0.843326 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - q^{19} + 12 q^{21} + 5 q^{22} - 8 q^{23} - q^{24} + 2 q^{26} - q^{27} + 2 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{32} - 15 q^{33} + 7 q^{34} + 14 q^{37} - q^{38} + 4 q^{39} + 17 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} - 8 q^{46} + 12 q^{47} - q^{48} + 11 q^{49} + 11 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 2 q^{56} + 9 q^{57} + 3 q^{58} + 16 q^{59} + 2 q^{61} + 4 q^{62} - 18 q^{63} + 3 q^{64} - 15 q^{66} + q^{67} + 7 q^{68} + 6 q^{69} + 8 q^{71} - 13 q^{73} + 14 q^{74} - q^{76} - 4 q^{77} + 4 q^{78} - 13 q^{81} + 17 q^{82} - 11 q^{83} + 12 q^{84} - 4 q^{86} - q^{87} + 5 q^{88} + 11 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{93} + 12 q^{94} - q^{96} + 24 q^{97} + 11 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 2 * q^7 + 3 * q^8 + 5 * q^11 - q^12 + 2 * q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - q^19 + 12 * q^21 + 5 * q^22 - 8 * q^23 - q^24 + 2 * q^26 - q^27 + 2 * q^28 + 3 * q^29 + 4 * q^31 + 3 * q^32 - 15 * q^33 + 7 * q^34 + 14 * q^37 - q^38 + 4 * q^39 + 17 * q^41 + 12 * q^42 - 4 * q^43 + 5 * q^44 - 8 * q^46 + 12 * q^47 - q^48 + 11 * q^49 + 11 * q^51 + 2 * q^52 - 6 * q^53 - q^54 + 2 * q^56 + 9 * q^57 + 3 * q^58 + 16 * q^59 + 2 * q^61 + 4 * q^62 - 18 * q^63 + 3 * q^64 - 15 * q^66 + q^67 + 7 * q^68 + 6 * q^69 + 8 * q^71 - 13 * q^73 + 14 * q^74 - q^76 - 4 * q^77 + 4 * q^78 - 13 * q^81 + 17 * q^82 - 11 * q^83 + 12 * q^84 - 4 * q^86 - q^87 + 5 * q^88 + 11 * q^89 - 4 * q^91 - 8 * q^92 - 24 * q^93 + 12 * q^94 - q^96 + 24 * q^97 + 11 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.81361	1.04709	0.523543	−	0.851999i	$-0.324610\pi$
$3$	1.81361	1.04709	0.523543	+	0.851999i	$0.324610\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.81361	0.740402
$7$	2.52444	0.954148	0.477074	−	0.878863i	$-0.341697\pi$
$7$	2.52444	0.954148	0.477074	+	0.878863i	$0.341697\pi$
$8$	1.00000	0.353553
$9$	0.289169	0.0963895
$10$	0	0
$11$	−2.91638	−0.879322	−0.439661	−	0.898164i	$-0.644901\pi$
$11$	−2.91638	−0.879322	−0.439661	+	0.898164i	$0.644901\pi$
$12$	1.81361	0.523543
$13$	3.10278	0.860555	0.430277	−	0.902697i	$-0.358416\pi$
$13$	3.10278	0.860555	0.430277	+	0.902697i	$0.358416\pi$
$14$	2.52444	0.674684
$15$	0	0
$16$	1.00000	0.250000
$17$	6.91638	1.67747	0.838734	−	0.544541i	$-0.183296\pi$
$17$	6.91638	1.67747	0.838734	+	0.544541i	$0.183296\pi$
$18$	0.289169	0.0681577
$19$	1.81361	0.416070	0.208035	−	0.978121i	$-0.433293\pi$
$19$	1.81361	0.416070	0.208035	+	0.978121i	$0.433293\pi$
$20$	0	0
$21$	4.57834	0.999075
$22$	−2.91638	−0.621775
$23$	−5.68111	−1.18459	−0.592297	−	0.805720i	$-0.701779\pi$
$23$	−5.68111	−1.18459	−0.592297	+	0.805720i	$0.701779\pi$
$24$	1.81361	0.370201
$25$	0	0
$26$	3.10278	0.608504
$27$	−4.91638	−0.946158
$28$	2.52444	0.477074
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.12193	−1.45874	−0.729371	−	0.684118i	$-0.760187\pi$
$31$	−8.12193	−1.45874	−0.729371	+	0.684118i	$0.760187\pi$
$32$	1.00000	0.176777
$33$	−5.28917	−0.920726
$34$	6.91638	1.18615
$35$	0	0
$36$	0.289169	0.0481948
$37$	7.39194	1.21523	0.607614	−	0.794232i	$-0.292127\pi$
$37$	7.39194	1.21523	0.607614	+	0.794232i	$0.292127\pi$
$38$	1.81361	0.294206
$39$	5.62721	0.901075
$40$	0	0
$41$	8.39194	1.31060	0.655301	−	0.755368i	$-0.272542\pi$
$41$	8.39194	1.31060	0.655301	+	0.755368i	$0.272542\pi$
$42$	4.57834	0.706453
$43$	−5.04888	−0.769946	−0.384973	−	0.922928i	$-0.625789\pi$
$43$	−5.04888	−0.769946	−0.384973	+	0.922928i	$0.625789\pi$
$44$	−2.91638	−0.439661
$45$	0	0
$46$	−5.68111	−0.837634
$47$	4.86751	0.709999	0.354999	−	0.934867i	$-0.384481\pi$
$47$	4.86751	0.709999	0.354999	+	0.934867i	$0.384481\pi$
$48$	1.81361	0.261772
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	12.5436	1.75645
$52$	3.10278	0.430277
$53$	5.30833	0.729155	0.364577	−	0.931173i	$-0.381214\pi$
$53$	5.30833	0.729155	0.364577	+	0.931173i	$0.381214\pi$
$54$	−4.91638	−0.669035
$55$	0	0
$56$	2.52444	0.337342
$57$	3.28917	0.435661
$58$	1.00000	0.131306
$59$	2.60806	0.339540	0.169770	−	0.985484i	$-0.445698\pi$
$59$	2.60806	0.339540	0.169770	+	0.985484i	$0.445698\pi$
$60$	0	0
$61$	−10.0680	−1.28908	−0.644540	−	0.764571i	$-0.722951\pi$
$61$	−10.0680	−1.28908	−0.644540	+	0.764571i	$0.722951\pi$
$62$	−8.12193	−1.03149
$63$	0.729988	0.0919699
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.28917	−0.651052
$67$	−2.68111	−0.327550	−0.163775	−	0.986498i	$-0.552367\pi$
$67$	−2.68111	−0.327550	−0.163775	+	0.986498i	$0.552367\pi$
$68$	6.91638	0.838734
$69$	−10.3033	−1.24037
$70$	0	0
$71$	5.68111	0.674224	0.337112	−	0.941465i	$-0.390550\pi$
$71$	5.68111	0.674224	0.337112	+	0.941465i	$0.390550\pi$
$72$	0.289169	0.0340788
$73$	−15.0680	−1.76358	−0.881790	−	0.471642i	$-0.843661\pi$
$73$	−15.0680	−1.76358	−0.881790	+	0.471642i	$0.843661\pi$
$74$	7.39194	0.859296
$75$	0	0
$76$	1.81361	0.208035
$77$	−7.36222	−0.839003
$78$	5.62721	0.637156
$79$	−1.73501	−0.195204	−0.0976020	−	0.995226i	$-0.531117\pi$
$79$	−1.73501	−0.195204	−0.0976020	+	0.995226i	$0.531117\pi$
$80$	0	0
$81$	−9.78389	−1.08710
$82$	8.39194	0.926735
$83$	−13.1219	−1.44032	−0.720160	−	0.693808i	$-0.755931\pi$
$83$	−13.1219	−1.44032	−0.720160	+	0.693808i	$0.755931\pi$
$84$	4.57834	0.499538
$85$	0	0
$86$	−5.04888	−0.544434
$87$	1.81361	0.194439
$88$	−2.91638	−0.310887
$89$	5.23527	0.554937	0.277469	−	0.960735i	$-0.410505\pi$
$89$	5.23527	0.554937	0.277469	+	0.960735i	$0.410505\pi$
$90$	0	0
$91$	7.83276	0.821097
$92$	−5.68111	−0.592297
$93$	−14.7300	−1.52743
$94$	4.86751	0.502045
$95$	0	0
$96$	1.81361	0.185100
$97$	1.27001	0.128950	0.0644751	−	0.997919i	$-0.479463\pi$
$97$	1.27001	0.128950	0.0644751	+	0.997919i	$0.479463\pi$
$98$	−0.627213	−0.0633581
$99$	−0.843326	−0.0847574
$100$	0	0
$101$	5.39194	0.536518	0.268259	−	0.963347i	$-0.413552\pi$
$101$	5.39194	0.536518	0.268259	+	0.963347i	$0.413552\pi$
$102$	12.5436	1.24200
$103$	−17.6116	−1.73533	−0.867663	−	0.497154i	$-0.834379\pi$
$103$	−17.6116	−1.73533	−0.867663	+	0.497154i	$0.834379\pi$
$104$	3.10278	0.304252
$105$	0	0
$106$	5.30833	0.515590
$107$	4.30833	0.416502	0.208251	−	0.978075i	$-0.433223\pi$
$107$	4.30833	0.416502	0.208251	+	0.978075i	$0.433223\pi$
$108$	−4.91638	−0.473079
$109$	−7.57331	−0.725392	−0.362696	−	0.931908i	$-0.618144\pi$
$109$	−7.57331	−0.725392	−0.362696	+	0.931908i	$0.618144\pi$
$110$	0	0
$111$	13.4061	1.27245
$112$	2.52444	0.238537
$113$	−5.60806	−0.527562	−0.263781	−	0.964583i	$-0.584970\pi$
$113$	−5.60806	−0.527562	−0.263781	+	0.964583i	$0.584970\pi$
$114$	3.28917	0.308059
$115$	0	0
$116$	1.00000	0.0928477
$117$	0.897225	0.0829485
$118$	2.60806	0.240091
$119$	17.4600	1.60055
$120$	0	0
$121$	−2.49472	−0.226793
$122$	−10.0680	−0.911517
$123$	15.2197	1.37231
$124$	−8.12193	−0.729371
$125$	0	0
$126$	0.729988	0.0650325
$127$	2.12193	0.188291	0.0941455	−	0.995558i	$-0.469988\pi$
$127$	2.12193	0.188291	0.0941455	+	0.995558i	$0.469988\pi$
$128$	1.00000	0.0883883
$129$	−9.15667	−0.806200
$130$	0	0
$131$	3.74055	0.326813	0.163407	−	0.986559i	$-0.447752\pi$
$131$	3.74055	0.326813	0.163407	+	0.986559i	$0.447752\pi$
$132$	−5.28917	−0.460363
$133$	4.57834	0.396992
$134$	−2.68111	−0.231613
$135$	0	0
$136$	6.91638	0.593075
$137$	11.7980	1.00797	0.503986	−	0.863712i	$-0.331866\pi$
$137$	11.7980	1.00797	0.503986	+	0.863712i	$0.331866\pi$
$138$	−10.3033	−0.877075
$139$	12.3083	1.04398	0.521989	−	0.852952i	$-0.325190\pi$
$139$	12.3083	1.04398	0.521989	+	0.852952i	$0.325190\pi$
$140$	0	0
$141$	8.82774	0.743430
$142$	5.68111	0.476748
$143$	−9.04888	−0.756705
$144$	0.289169	0.0240974
$145$	0	0
$146$	−15.0680	−1.24704
$147$	−1.13752	−0.0938209
$148$	7.39194	0.607614
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−8.89722	−0.724046	−0.362023	−	0.932169i	$-0.617914\pi$
$151$	−8.89722	−0.724046	−0.362023	+	0.932169i	$0.617914\pi$
$152$	1.81361	0.147103
$153$	2.00000	0.161690
$154$	−7.36222	−0.593265
$155$	0	0
$156$	5.62721	0.450538
$157$	−17.4897	−1.39583	−0.697915	−	0.716181i	$-0.745889\pi$
$157$	−17.4897	−1.39583	−0.697915	+	0.716181i	$0.745889\pi$
$158$	−1.73501	−0.138030
$159$	9.62721	0.763488
$160$	0	0
$161$	−14.3416	−1.13028
$162$	−9.78389	−0.768695
$163$	−3.02972	−0.237306	−0.118653	−	0.992936i	$-0.537858\pi$
$163$	−3.02972	−0.237306	−0.118653	+	0.992936i	$0.537858\pi$
$164$	8.39194	0.655301
$165$	0	0
$166$	−13.1219	−1.01846
$167$	−23.9305	−1.85180	−0.925899	−	0.377770i	$-0.876691\pi$
$167$	−23.9305	−1.85180	−0.925899	+	0.377770i	$0.876691\pi$
$168$	4.57834	0.353226
$169$	−3.37279	−0.259445
$170$	0	0
$171$	0.524438	0.0401048
$172$	−5.04888	−0.384973
$173$	11.5194	0.875805	0.437902	−	0.899022i	$-0.355722\pi$
$173$	11.5194	0.875805	0.437902	+	0.899022i	$0.355722\pi$
$174$	1.81361	0.137489
$175$	0	0
$176$	−2.91638	−0.219831
$177$	4.72999	0.355528
$178$	5.23527	0.392400
$179$	8.30833	0.620993	0.310497	−	0.950574i	$-0.399505\pi$
$179$	8.30833	0.620993	0.310497	+	0.950574i	$0.399505\pi$
$180$	0	0
$181$	17.7194	1.31707	0.658537	−	0.752548i	$-0.271175\pi$
$181$	17.7194	1.31707	0.658537	+	0.752548i	$0.271175\pi$
$182$	7.83276	0.580603
$183$	−18.2594	−1.34978
$184$	−5.68111	−0.418817
$185$	0	0
$186$	−14.7300	−1.08006
$187$	−20.1708	−1.47504
$188$	4.86751	0.354999
$189$	−12.4111	−0.902775
$190$	0	0
$191$	17.1708	1.24244	0.621218	−	0.783638i	$-0.286638\pi$
$191$	17.1708	1.24244	0.621218	+	0.783638i	$0.286638\pi$
$192$	1.81361	0.130886
$193$	7.06803	0.508768	0.254384	−	0.967103i	$-0.418127\pi$
$193$	7.06803	0.508768	0.254384	+	0.967103i	$0.418127\pi$
$194$	1.27001	0.0911815
$195$	0	0
$196$	−0.627213	−0.0448009
$197$	−16.1361	−1.14965	−0.574824	−	0.818277i	$-0.694929\pi$
$197$	−16.1361	−1.14965	−0.574824	+	0.818277i	$0.694929\pi$
$198$	−0.843326	−0.0599326
$199$	8.25945	0.585497	0.292748	−	0.956190i	$-0.405430\pi$
$199$	8.25945	0.585497	0.292748	+	0.956190i	$0.405430\pi$
$200$	0	0
$201$	−4.86248	−0.342973
$202$	5.39194	0.379376
$203$	2.52444	0.177181
$204$	12.5436	0.878227
$205$	0	0
$206$	−17.6116	−1.22706
$207$	−1.64280	−0.114182
$208$	3.10278	0.215139
$209$	−5.28917	−0.365859
$210$	0	0
$211$	−12.2302	−0.841965	−0.420982	−	0.907069i	$-0.638315\pi$
$211$	−12.2302	−0.841965	−0.420982	+	0.907069i	$0.638315\pi$
$212$	5.30833	0.364577
$213$	10.3033	0.705971
$214$	4.30833	0.294511
$215$	0	0
$216$	−4.91638	−0.334517
$217$	−20.5033	−1.39186
$218$	−7.57331	−0.512930
$219$	−27.3275	−1.84662
$220$	0	0
$221$	21.4600	1.44355
$222$	13.4061	0.899757
$223$	11.2544	0.753652	0.376826	−	0.926284i	$-0.377015\pi$
$223$	11.2544	0.753652	0.376826	+	0.926284i	$0.377015\pi$
$224$	2.52444	0.168671
$225$	0	0
$226$	−5.60806	−0.373042
$227$	−19.7647	−1.31183	−0.655916	−	0.754834i	$-0.727717\pi$
$227$	−19.7647	−1.31183	−0.655916	+	0.754834i	$0.727717\pi$
$228$	3.28917	0.217831
$229$	−23.7250	−1.56779	−0.783895	−	0.620894i	$-0.786770\pi$
$229$	−23.7250	−1.56779	−0.783895	+	0.620894i	$0.786770\pi$
$230$	0	0
$231$	−13.3522	−0.878509
$232$	1.00000	0.0656532
$233$	24.0625	1.57639	0.788193	−	0.615428i	$-0.211017\pi$
$233$	24.0625	1.57639	0.788193	+	0.615428i	$0.211017\pi$
$234$	0.897225	0.0586534
$235$	0	0
$236$	2.60806	0.169770
$237$	−3.14663	−0.204395
$238$	17.4600	1.13176
$239$	−4.31335	−0.279007	−0.139504	−	0.990222i	$-0.544551\pi$
$239$	−4.31335	−0.279007	−0.139504	+	0.990222i	$0.544551\pi$
$240$	0	0
$241$	−27.7144	−1.78524	−0.892621	−	0.450808i	$-0.851136\pi$
$241$	−27.7144	−1.78524	−0.892621	+	0.450808i	$0.851136\pi$
$242$	−2.49472	−0.160367
$243$	−2.99498	−0.192128
$244$	−10.0680	−0.644540
$245$	0	0
$246$	15.2197	0.970372
$247$	5.62721	0.358051
$248$	−8.12193	−0.515743
$249$	−23.7980	−1.50814
$250$	0	0
$251$	−21.7542	−1.37311	−0.686555	−	0.727077i	$-0.740878\pi$
$251$	−21.7542	−1.37311	−0.686555	+	0.727077i	$0.740878\pi$
$252$	0.729988	0.0459849
$253$	16.5683	1.04164
$254$	2.12193	0.133142
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.83276	0.363838	0.181919	−	0.983314i	$-0.441769\pi$
$257$	5.83276	0.363838	0.181919	+	0.983314i	$0.441769\pi$
$258$	−9.15667	−0.570070
$259$	18.6605	1.15951
$260$	0	0
$261$	0.289169	0.0178991
$262$	3.74055	0.231092
$263$	20.8675	1.28675	0.643373	−	0.765553i	$-0.277535\pi$
$263$	20.8675	1.28675	0.643373	+	0.765553i	$0.277535\pi$
$264$	−5.28917	−0.325526
$265$	0	0
$266$	4.57834	0.280716
$267$	9.49472	0.581067
$268$	−2.68111	−0.163775
$269$	0.548618	0.0334498	0.0167249	−	0.999860i	$-0.494676\pi$
$269$	0.548618	0.0334498	0.0167249	+	0.999860i	$0.494676\pi$
$270$	0	0
$271$	5.00357	0.303945	0.151973	−	0.988385i	$-0.451437\pi$
$271$	5.00357	0.303945	0.151973	+	0.988385i	$0.451437\pi$
$272$	6.91638	0.419367
$273$	14.2056	0.859759
$274$	11.7980	0.712744
$275$	0	0
$276$	−10.3033	−0.620186
$277$	0.632236	0.0379874	0.0189937	−	0.999820i	$-0.493954\pi$
$277$	0.632236	0.0379874	0.0189937	+	0.999820i	$0.493954\pi$
$278$	12.3083	0.738204
$279$	−2.34861	−0.140607
$280$	0	0
$281$	30.9653	1.84723	0.923616	−	0.383319i	$-0.125219\pi$
$281$	30.9653	1.84723	0.923616	+	0.383319i	$0.125219\pi$
$282$	8.82774	0.525684
$283$	−6.33804	−0.376758	−0.188379	−	0.982096i	$-0.560323\pi$
$283$	−6.33804	−0.376758	−0.188379	+	0.982096i	$0.560323\pi$
$284$	5.68111	0.337112
$285$	0	0
$286$	−9.04888	−0.535071
$287$	21.1849	1.25051
$288$	0.289169	0.0170394
$289$	30.8363	1.81390
$290$	0	0
$291$	2.30330	0.135022
$292$	−15.0680	−0.881790
$293$	−28.4197	−1.66030	−0.830148	−	0.557543i	$-0.811744\pi$
$293$	−28.4197	−1.66030	−0.830148	+	0.557543i	$0.811744\pi$
$294$	−1.13752	−0.0663414
$295$	0	0
$296$	7.39194	0.429648
$297$	14.3380	0.831978
$298$	2.00000	0.115857
$299$	−17.6272	−1.01941
$300$	0	0
$301$	−12.7456	−0.734643
$302$	−8.89722	−0.511978
$303$	9.77886	0.561781
$304$	1.81361	0.104017
$305$	0	0
$306$	2.00000	0.114332
$307$	22.6797	1.29440	0.647198	−	0.762322i	$-0.275941\pi$
$307$	22.6797	1.29440	0.647198	+	0.762322i	$0.275941\pi$
$308$	−7.36222	−0.419502
$309$	−31.9406	−1.81704
$310$	0	0
$311$	−9.08719	−0.515287	−0.257644	−	0.966240i	$-0.582946\pi$
$311$	−9.08719	−0.515287	−0.257644	+	0.966240i	$0.582946\pi$
$312$	5.62721	0.318578
$313$	25.0630	1.41665	0.708323	−	0.705889i	$-0.249452\pi$
$313$	25.0630	1.41665	0.708323	+	0.705889i	$0.249452\pi$
$314$	−17.4897	−0.987001
$315$	0	0
$316$	−1.73501	−0.0976020
$317$	−14.9497	−0.839657	−0.419829	−	0.907603i	$-0.637910\pi$
$317$	−14.9497	−0.839657	−0.419829	+	0.907603i	$0.637910\pi$
$318$	9.62721	0.539867
$319$	−2.91638	−0.163286
$320$	0	0
$321$	7.81361	0.436113
$322$	−14.3416	−0.799227
$323$	12.5436	0.697944
$324$	−9.78389	−0.543549
$325$	0	0
$326$	−3.02972	−0.167801
$327$	−13.7350	−0.759548
$328$	8.39194	0.463368
$329$	12.2877	0.677444
$330$	0	0
$331$	−18.1270	−0.996348	−0.498174	−	0.867077i	$-0.665996\pi$
$331$	−18.1270	−0.996348	−0.498174	+	0.867077i	$0.665996\pi$
$332$	−13.1219	−0.720160
$333$	2.13752	0.117135
$334$	−23.9305	−1.30942
$335$	0	0
$336$	4.57834	0.249769
$337$	9.08362	0.494816	0.247408	−	0.968911i	$-0.420421\pi$
$337$	9.08362	0.494816	0.247408	+	0.968911i	$0.420421\pi$
$338$	−3.37279	−0.183455
$339$	−10.1708	−0.552402
$340$	0	0
$341$	23.6867	1.28270
$342$	0.524438	0.0283584
$343$	−19.2544	−1.03964
$344$	−5.04888	−0.272217
$345$	0	0
$346$	11.5194	0.619288
$347$	−30.7194	−1.64911	−0.824553	−	0.565785i	$-0.808573\pi$
$347$	−30.7194	−1.64911	−0.824553	+	0.565785i	$0.808573\pi$
$348$	1.81361	0.0972195
$349$	15.7789	0.844623	0.422312	−	0.906451i	$-0.361219\pi$
$349$	15.7789	0.844623	0.422312	+	0.906451i	$0.361219\pi$
$350$	0	0
$351$	−15.2544	−0.814221
$352$	−2.91638	−0.155444
$353$	−31.6797	−1.68614	−0.843069	−	0.537805i	$-0.819254\pi$
$353$	−31.6797	−1.68614	−0.843069	+	0.537805i	$0.819254\pi$
$354$	4.72999	0.251396
$355$	0	0
$356$	5.23527	0.277469
$357$	31.6655	1.67592
$358$	8.30833	0.439109
$359$	−27.1013	−1.43035	−0.715177	−	0.698944i	$-0.753654\pi$
$359$	−27.1013	−1.43035	−0.715177	+	0.698944i	$0.753654\pi$
$360$	0	0
$361$	−15.7108	−0.826886
$362$	17.7194	0.931312
$363$	−4.52444	−0.237471
$364$	7.83276	0.410548
$365$	0	0
$366$	−18.2594	−0.954437
$367$	31.8953	1.66492	0.832459	−	0.554086i	$-0.186932\pi$
$367$	31.8953	1.66492	0.832459	+	0.554086i	$0.186932\pi$
$368$	−5.68111	−0.296148
$369$	2.42669	0.126328
$370$	0	0
$371$	13.4005	0.695721
$372$	−14.7300	−0.763714
$373$	28.0383	1.45177	0.725884	−	0.687817i	$-0.241431\pi$
$373$	28.0383	1.45177	0.725884	+	0.687817i	$0.241431\pi$
$374$	−20.1708	−1.04301
$375$	0	0
$376$	4.86751	0.251022
$377$	3.10278	0.159801
$378$	−12.4111	−0.638358
$379$	8.69525	0.446645	0.223322	−	0.974745i	$-0.428310\pi$
$379$	8.69525	0.446645	0.223322	+	0.974745i	$0.428310\pi$
$380$	0	0
$381$	3.84835	0.197157
$382$	17.1708	0.878535
$383$	14.8917	0.760930	0.380465	−	0.924795i	$-0.375764\pi$
$383$	14.8917	0.760930	0.380465	+	0.924795i	$0.375764\pi$
$384$	1.81361	0.0925502
$385$	0	0
$386$	7.06803	0.359753
$387$	−1.45998	−0.0742148
$388$	1.27001	0.0644751
$389$	−1.93197	−0.0979546	−0.0489773	−	0.998800i	$-0.515596\pi$
$389$	−1.93197	−0.0979546	−0.0489773	+	0.998800i	$0.515596\pi$
$390$	0	0
$391$	−39.2927	−1.98712
$392$	−0.627213	−0.0316790
$393$	6.78389	0.342202
$394$	−16.1361	−0.812923
$395$	0	0
$396$	−0.843326	−0.0423787
$397$	19.2983	0.968553	0.484276	−	0.874915i	$-0.339083\pi$
$397$	19.2983	0.968553	0.484276	+	0.874915i	$0.339083\pi$
$398$	8.25945	0.414009
$399$	8.30330	0.415685
$400$	0	0
$401$	16.7250	0.835205	0.417602	−	0.908630i	$-0.362870\pi$
$401$	16.7250	0.835205	0.417602	+	0.908630i	$0.362870\pi$
$402$	−4.86248	−0.242519
$403$	−25.2005	−1.25533
$404$	5.39194	0.268259
$405$	0	0
$406$	2.52444	0.125286
$407$	−21.5577	−1.06858
$408$	12.5436	0.621000
$409$	−26.4842	−1.30956	−0.654779	−	0.755821i	$-0.727238\pi$
$409$	−26.4842	−1.30956	−0.654779	+	0.755821i	$0.727238\pi$
$410$	0	0
$411$	21.3970	1.05543
$412$	−17.6116	−0.867663
$413$	6.58388	0.323971
$414$	−1.64280	−0.0807392
$415$	0	0
$416$	3.10278	0.152126
$417$	22.3225	1.09314
$418$	−5.28917	−0.258702
$419$	−6.94610	−0.339339	−0.169670	−	0.985501i	$-0.554270\pi$
$419$	−6.94610	−0.339339	−0.169670	+	0.985501i	$0.554270\pi$
$420$	0	0
$421$	−2.71585	−0.132363	−0.0661813	−	0.997808i	$-0.521082\pi$
$421$	−2.71585	−0.132363	−0.0661813	+	0.997808i	$0.521082\pi$
$422$	−12.2302	−0.595359
$423$	1.40753	0.0684364
$424$	5.30833	0.257795
$425$	0	0
$426$	10.3033	0.499197
$427$	−25.4161	−1.22997
$428$	4.30833	0.208251
$429$	−16.4111	−0.792335
$430$	0	0
$431$	16.2978	0.785036	0.392518	−	0.919744i	$-0.371604\pi$
$431$	16.2978	0.785036	0.392518	+	0.919744i	$0.371604\pi$
$432$	−4.91638	−0.236540
$433$	−10.1169	−0.486188	−0.243094	−	0.970003i	$-0.578162\pi$
$433$	−10.1169	−0.486188	−0.243094	+	0.970003i	$0.578162\pi$
$434$	−20.5033	−0.984190
$435$	0	0
$436$	−7.57331	−0.362696
$437$	−10.3033	−0.492874
$438$	−27.3275	−1.30576
$439$	9.79445	0.467464	0.233732	−	0.972301i	$-0.424906\pi$
$439$	9.79445	0.467464	0.233732	+	0.972301i	$0.424906\pi$
$440$	0	0
$441$	−0.181370	−0.00863668
$442$	21.4600	1.02075
$443$	−21.7980	−1.03566	−0.517828	−	0.855485i	$-0.673259\pi$
$443$	−21.7980	−1.03566	−0.517828	+	0.855485i	$0.673259\pi$
$444$	13.4061	0.636224
$445$	0	0
$446$	11.2544	0.532913
$447$	3.62721	0.171561
$448$	2.52444	0.119268
$449$	−22.2686	−1.05092	−0.525459	−	0.850819i	$-0.676106\pi$
$449$	−22.2686	−1.05092	−0.525459	+	0.850819i	$0.676106\pi$
$450$	0	0
$451$	−24.4741	−1.15244
$452$	−5.60806	−0.263781
$453$	−16.1361	−0.758138
$454$	−19.7647	−0.927605
$455$	0	0
$456$	3.28917	0.154029
$457$	−29.8958	−1.39847	−0.699233	−	0.714894i	$-0.746475\pi$
$457$	−29.8958	−1.39847	−0.699233	+	0.714894i	$0.746475\pi$
$458$	−23.7250	−1.10859
$459$	−34.0036	−1.58715
$460$	0	0
$461$	33.2927	1.55060	0.775299	−	0.631595i	$-0.217599\pi$
$461$	33.2927	1.55060	0.775299	+	0.631595i	$0.217599\pi$
$462$	−13.3522	−0.621200
$463$	13.1184	0.609662	0.304831	−	0.952406i	$-0.401400\pi$
$463$	13.1184	0.609662	0.304831	+	0.952406i	$0.401400\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	24.0625	1.11467
$467$	−4.15165	−0.192115	−0.0960577	−	0.995376i	$-0.530623\pi$
$467$	−4.15165	−0.192115	−0.0960577	+	0.995376i	$0.530623\pi$
$468$	0.897225	0.0414742
$469$	−6.76830	−0.312531
$470$	0	0
$471$	−31.7194	−1.46155
$472$	2.60806	0.120046
$473$	14.7244	0.677031
$474$	−3.14663	−0.144529
$475$	0	0
$476$	17.4600	0.800277
$477$	1.53500	0.0702829
$478$	−4.31335	−0.197288
$479$	42.7144	1.95167	0.975835	−	0.218507i	$-0.0701186\pi$
$479$	42.7144	1.95167	0.975835	+	0.218507i	$0.0701186\pi$
$480$	0	0
$481$	22.9355	1.04577
$482$	−27.7144	−1.26236
$483$	−26.0100	−1.18350
$484$	−2.49472	−0.113396
$485$	0	0
$486$	−2.99498	−0.135855
$487$	18.3728	0.832550	0.416275	−	0.909239i	$-0.363335\pi$
$487$	18.3728	0.832550	0.416275	+	0.909239i	$0.363335\pi$
$488$	−10.0680	−0.455758
$489$	−5.49472	−0.248480
$490$	0	0
$491$	−0.881639	−0.0397878	−0.0198939	−	0.999802i	$-0.506333\pi$
$491$	−0.881639	−0.0397878	−0.0198939	+	0.999802i	$0.506333\pi$
$492$	15.2197	0.686156
$493$	6.91638	0.311498
$494$	5.62721	0.253180
$495$	0	0
$496$	−8.12193	−0.364685
$497$	14.3416	0.643309
$498$	−23.7980	−1.06641
$499$	32.4791	1.45397	0.726983	−	0.686656i	$-0.240922\pi$
$499$	32.4791	1.45397	0.726983	+	0.686656i	$0.240922\pi$
$500$	0	0
$501$	−43.4005	−1.93899
$502$	−21.7542	−0.970936
$503$	32.6620	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$503$	32.6620	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$504$	0.729988	0.0325163
$505$	0	0
$506$	16.5683	0.736550
$507$	−6.11691	−0.271661
$508$	2.12193	0.0941455
$509$	27.3311	1.21143	0.605714	−	0.795683i	$-0.292888\pi$
$509$	27.3311	1.21143	0.605714	+	0.795683i	$0.292888\pi$
$510$	0	0
$511$	−38.0383	−1.68272
$512$	1.00000	0.0441942
$513$	−8.91638	−0.393668
$514$	5.83276	0.257272
$515$	0	0
$516$	−9.15667	−0.403100
$517$	−14.1955	−0.624318
$518$	18.6605	0.819895
$519$	20.8917	0.917043
$520$	0	0
$521$	11.5400	0.505578	0.252789	−	0.967521i	$-0.418652\pi$
$521$	11.5400	0.505578	0.252789	+	0.967521i	$0.418652\pi$
$522$	0.289169	0.0126566
$523$	−19.4056	−0.848546	−0.424273	−	0.905534i	$-0.639470\pi$
$523$	−19.4056	−0.848546	−0.424273	+	0.905534i	$0.639470\pi$
$524$	3.74055	0.163407
$525$	0	0
$526$	20.8675	0.909866
$527$	−56.1744	−2.44699
$528$	−5.28917	−0.230182
$529$	9.27504	0.403262
$530$	0	0
$531$	0.754168	0.0327281
$532$	4.57834	0.198496
$533$	26.0383	1.12784
$534$	9.49472	0.410877
$535$	0	0
$536$	−2.68111	−0.115806
$537$	15.0680	0.650234
$538$	0.548618	0.0236526
$539$	1.82919	0.0787889
$540$	0	0
$541$	26.7230	1.14891	0.574456	−	0.818536i	$-0.305214\pi$
$541$	26.7230	1.14891	0.574456	+	0.818536i	$0.305214\pi$
$542$	5.00357	0.214922
$543$	32.1361	1.37909
$544$	6.91638	0.296537
$545$	0	0
$546$	14.2056	0.607941
$547$	20.7875	0.888808	0.444404	−	0.895827i	$-0.353416\pi$
$547$	20.7875	0.888808	0.444404	+	0.895827i	$0.353416\pi$
$548$	11.7980	0.503986
$549$	−2.91136	−0.124254
$550$	0	0
$551$	1.81361	0.0772622
$552$	−10.3033	−0.438538
$553$	−4.37993	−0.186253
$554$	0.632236	0.0268611
$555$	0	0
$556$	12.3083	0.521989
$557$	11.0489	0.468156	0.234078	−	0.972218i	$-0.424793\pi$
$557$	11.0489	0.468156	0.234078	+	0.972218i	$0.424793\pi$
$558$	−2.34861	−0.0994245
$559$	−15.6655	−0.662581
$560$	0	0
$561$	−36.5819	−1.54449
$562$	30.9653	1.30619
$563$	−27.3311	−1.15187	−0.575933	−	0.817497i	$-0.695361\pi$
$563$	−27.3311	−1.15187	−0.575933	+	0.817497i	$0.695361\pi$
$564$	8.82774	0.371715
$565$	0	0
$566$	−6.33804	−0.266408
$567$	−24.6988	−1.03725
$568$	5.68111	0.238374
$569$	−6.54359	−0.274322	−0.137161	−	0.990549i	$-0.543798\pi$
$569$	−6.54359	−0.274322	−0.137161	+	0.990549i	$0.543798\pi$
$570$	0	0
$571$	46.4691	1.94467	0.972335	−	0.233589i	$-0.0750471\pi$
$571$	46.4691	1.94467	0.972335	+	0.233589i	$0.0750471\pi$
$572$	−9.04888	−0.378353
$573$	31.1411	1.30094
$574$	21.1849	0.884242
$575$	0	0
$576$	0.289169	0.0120487
$577$	40.4585	1.68431	0.842155	−	0.539235i	$-0.181287\pi$
$577$	40.4585	1.68431	0.842155	+	0.539235i	$0.181287\pi$
$578$	30.8363	1.28262
$579$	12.8186	0.532724
$580$	0	0
$581$	−33.1255	−1.37428
$582$	2.30330	0.0954749
$583$	−15.4811	−0.641162
$584$	−15.0680	−0.623520
$585$	0	0
$586$	−28.4197	−1.17401
$587$	15.1900	0.626957	0.313478	−	0.949595i	$-0.398506\pi$
$587$	15.1900	0.626957	0.313478	+	0.949595i	$0.398506\pi$
$588$	−1.13752	−0.0469104
$589$	−14.7300	−0.606939
$590$	0	0
$591$	−29.2645	−1.20378
$592$	7.39194	0.303807
$593$	−2.21611	−0.0910048	−0.0455024	−	0.998964i	$-0.514489\pi$
$593$	−2.21611	−0.0910048	−0.0455024	+	0.998964i	$0.514489\pi$
$594$	14.3380	0.588297
$595$	0	0
$596$	2.00000	0.0819232
$597$	14.9794	0.613066
$598$	−17.6272	−0.720830
$599$	−39.8852	−1.62967	−0.814833	−	0.579696i	$-0.803171\pi$
$599$	−39.8852	−1.62967	−0.814833	+	0.579696i	$0.803171\pi$
$600$	0	0
$601$	20.6897	0.843951	0.421975	−	0.906607i	$-0.361337\pi$
$601$	20.6897	0.843951	0.421975	+	0.906607i	$0.361337\pi$
$602$	−12.7456	−0.519471
$603$	−0.775293	−0.0315724
$604$	−8.89722	−0.362023
$605$	0	0
$606$	9.77886	0.397239
$607$	−16.6025	−0.673875	−0.336938	−	0.941527i	$-0.609391\pi$
$607$	−16.6025	−0.673875	−0.336938	+	0.941527i	$0.609391\pi$
$608$	1.81361	0.0735515
$609$	4.57834	0.185524
$610$	0	0
$611$	15.1028	0.610993
$612$	2.00000	0.0808452
$613$	8.13607	0.328613	0.164306	−	0.986409i	$-0.447461\pi$
$613$	8.13607	0.328613	0.164306	+	0.986409i	$0.447461\pi$
$614$	22.6797	0.915277
$615$	0	0
$616$	−7.36222	−0.296632
$617$	36.0822	1.45261	0.726307	−	0.687371i	$-0.241235\pi$
$617$	36.0822	1.45261	0.726307	+	0.687371i	$0.241235\pi$
$618$	−31.9406	−1.28484
$619$	13.9617	0.561168	0.280584	−	0.959830i	$-0.409472\pi$
$619$	13.9617	0.561168	0.280584	+	0.959830i	$0.409472\pi$
$620$	0	0
$621$	27.9305	1.12081
$622$	−9.08719	−0.364363
$623$	13.2161	0.529492
$624$	5.62721	0.225269
$625$	0	0
$626$	25.0630	1.00172
$627$	−9.59247	−0.383086
$628$	−17.4897	−0.697915
$629$	51.1255	2.03851
$630$	0	0
$631$	−5.49829	−0.218883	−0.109442	−	0.993993i	$-0.534906\pi$
$631$	−5.49829	−0.218883	−0.109442	+	0.993993i	$0.534906\pi$
$632$	−1.73501	−0.0690150
$633$	−22.1809	−0.881610
$634$	−14.9497	−0.593727
$635$	0	0
$636$	9.62721	0.381744
$637$	−1.94610	−0.0771073
$638$	−2.91638	−0.115461
$639$	1.64280	0.0649881
$640$	0	0
$641$	18.5033	0.730837	0.365418	−	0.930843i	$-0.380926\pi$
$641$	18.5033	0.730837	0.365418	+	0.930843i	$0.380926\pi$
$642$	7.81361	0.308378
$643$	25.3608	1.00013	0.500066	−	0.865988i	$-0.333309\pi$
$643$	25.3608	1.00013	0.500066	+	0.865988i	$0.333309\pi$
$644$	−14.3416	−0.565139
$645$	0	0
$646$	12.5436	0.493521
$647$	36.6761	1.44189	0.720943	−	0.692994i	$-0.243709\pi$
$647$	36.6761	1.44189	0.720943	+	0.692994i	$0.243709\pi$
$648$	−9.78389	−0.384347
$649$	−7.60609	−0.298565
$650$	0	0
$651$	−37.1849	−1.45739
$652$	−3.02972	−0.118653
$653$	4.47913	0.175282	0.0876410	−	0.996152i	$-0.472067\pi$
$653$	4.47913	0.175282	0.0876410	+	0.996152i	$0.472067\pi$
$654$	−13.7350	−0.537081
$655$	0	0
$656$	8.39194	0.327650
$657$	−4.35720	−0.169991
$658$	12.2877	0.479025
$659$	−16.0630	−0.625726	−0.312863	−	0.949798i	$-0.601288\pi$
$659$	−16.0630	−0.625726	−0.312863	+	0.949798i	$0.601288\pi$
$660$	0	0
$661$	−22.5244	−0.876099	−0.438050	−	0.898951i	$-0.644331\pi$
$661$	−22.5244	−0.876099	−0.438050	+	0.898951i	$0.644331\pi$
$662$	−18.1270	−0.704524
$663$	38.9200	1.51153
$664$	−13.1219	−0.509230
$665$	0	0
$666$	2.13752	0.0828271
$667$	−5.68111	−0.219974
$668$	−23.9305	−0.925899
$669$	20.4111	0.789139
$670$	0	0
$671$	29.3622	1.13352
$672$	4.57834	0.176613
$673$	−41.2474	−1.58997	−0.794986	−	0.606628i	$-0.792522\pi$
$673$	−41.2474	−1.58997	−0.794986	+	0.606628i	$0.792522\pi$
$674$	9.08362	0.349888
$675$	0	0
$676$	−3.37279	−0.129723
$677$	17.1653	0.659715	0.329857	−	0.944031i	$-0.392999\pi$
$677$	17.1653	0.659715	0.329857	+	0.944031i	$0.392999\pi$
$678$	−10.1708	−0.390608
$679$	3.20607	0.123038
$680$	0	0
$681$	−35.8454	−1.37360
$682$	23.6867	0.907009
$683$	−20.2786	−0.775939	−0.387970	−	0.921672i	$-0.626823\pi$
$683$	−20.2786	−0.775939	−0.387970	+	0.921672i	$0.626823\pi$
$684$	0.524438	0.0200524
$685$	0	0
$686$	−19.2544	−0.735137
$687$	−43.0278	−1.64161
$688$	−5.04888	−0.192487
$689$	16.4705	0.627478
$690$	0	0
$691$	22.2106	0.844930	0.422465	−	0.906379i	$-0.361165\pi$
$691$	22.2106	0.844930	0.422465	+	0.906379i	$0.361165\pi$
$692$	11.5194	0.437902
$693$	−2.12892	−0.0808711
$694$	−30.7194	−1.16609
$695$	0	0
$696$	1.81361	0.0687446
$697$	58.0419	2.19849
$698$	15.7789	0.597239
$699$	43.6399	1.65061
$700$	0	0
$701$	−11.4005	−0.430592	−0.215296	−	0.976549i	$-0.569072\pi$
$701$	−11.4005	−0.430592	−0.215296	+	0.976549i	$0.569072\pi$
$702$	−15.2544	−0.575741
$703$	13.4061	0.505620
$704$	−2.91638	−0.109915
$705$	0	0
$706$	−31.6797	−1.19228
$707$	13.6116	0.511918
$708$	4.72999	0.177764
$709$	37.3311	1.40200	0.700999	−	0.713163i	$-0.252738\pi$
$709$	37.3311	1.40200	0.700999	+	0.713163i	$0.252738\pi$
$710$	0	0
$711$	−0.501711	−0.0188156
$712$	5.23527	0.196200
$713$	46.1416	1.72802
$714$	31.6655	1.18505
$715$	0	0
$716$	8.30833	0.310497
$717$	−7.82272	−0.292145
$718$	−27.1013	−1.01141
$719$	18.5839	0.693062	0.346531	−	0.938039i	$-0.387360\pi$
$719$	18.5839	0.693062	0.346531	+	0.938039i	$0.387360\pi$
$720$	0	0
$721$	−44.4595	−1.65576
$722$	−15.7108	−0.584697
$723$	−50.2630	−1.86930
$724$	17.7194	0.658537
$725$	0	0
$726$	−4.52444	−0.167918
$727$	−29.8227	−1.10606	−0.553032	−	0.833160i	$-0.686529\pi$
$727$	−29.8227	−1.10606	−0.553032	+	0.833160i	$0.686529\pi$
$728$	7.83276	0.290302
$729$	23.9200	0.885924
$730$	0	0
$731$	−34.9200	−1.29156
$732$	−18.2594	−0.674889
$733$	−32.2056	−1.18954	−0.594770	−	0.803896i	$-0.702757\pi$
$733$	−32.2056	−1.18954	−0.594770	+	0.803896i	$0.702757\pi$
$734$	31.8953	1.17728
$735$	0	0
$736$	−5.68111	−0.209409
$737$	7.81915	0.288022
$738$	2.42669	0.0893276
$739$	27.8328	1.02384	0.511922	−	0.859032i	$-0.328934\pi$
$739$	27.8328	1.02384	0.511922	+	0.859032i	$0.328934\pi$
$740$	0	0
$741$	10.2056	0.374910
$742$	13.4005	0.491949
$743$	31.4288	1.15301	0.576506	−	0.817093i	$-0.304416\pi$
$743$	31.4288	1.15301	0.576506	+	0.817093i	$0.304416\pi$
$744$	−14.7300	−0.540028
$745$	0	0
$746$	28.0383	1.02656
$747$	−3.79445	−0.138832
$748$	−20.1708	−0.737518
$749$	10.8761	0.397404
$750$	0	0
$751$	14.4635	0.527782	0.263891	−	0.964552i	$-0.414994\pi$
$751$	14.4635	0.527782	0.263891	+	0.964552i	$0.414994\pi$
$752$	4.86751	0.177500
$753$	−39.4535	−1.43777
$754$	3.10278	0.112996
$755$	0	0
$756$	−12.4111	−0.451387
$757$	−34.0071	−1.23601	−0.618005	−	0.786174i	$-0.712059\pi$
$757$	−34.0071	−1.23601	−0.618005	+	0.786174i	$0.712059\pi$
$758$	8.69525	0.315826
$759$	30.0484	1.09069
$760$	0	0
$761$	13.0594	0.473404	0.236702	−	0.971582i	$-0.423933\pi$
$761$	13.0594	0.473404	0.236702	+	0.971582i	$0.423933\pi$
$762$	3.84835	0.139411
$763$	−19.1184	−0.692131
$764$	17.1708	0.621218
$765$	0	0
$766$	14.8917	0.538058
$767$	8.09221	0.292193
$768$	1.81361	0.0654429
$769$	47.2525	1.70397	0.851984	−	0.523568i	$-0.175400\pi$
$769$	47.2525	1.70397	0.851984	+	0.523568i	$0.175400\pi$
$770$	0	0
$771$	10.5783	0.380970
$772$	7.06803	0.254384
$773$	−12.6675	−0.455618	−0.227809	−	0.973706i	$-0.573156\pi$
$773$	−12.6675	−0.455618	−0.227809	+	0.973706i	$0.573156\pi$
$774$	−1.45998	−0.0524778
$775$	0	0
$776$	1.27001	0.0455908
$777$	33.8428	1.21410
$778$	−1.93197	−0.0692644
$779$	15.2197	0.545302
$780$	0	0
$781$	−16.5683	−0.592860
$782$	−39.2927	−1.40511
$783$	−4.91638	−0.175697
$784$	−0.627213	−0.0224005
$785$	0	0
$786$	6.78389	0.241973
$787$	−30.0297	−1.07044	−0.535222	−	0.844711i	$-0.679772\pi$
$787$	−30.0297	−1.07044	−0.535222	+	0.844711i	$0.679772\pi$
$788$	−16.1361	−0.574824
$789$	37.8454	1.34733
$790$	0	0
$791$	−14.1572	−0.503372
$792$	−0.843326	−0.0299663
$793$	−31.2388	−1.10932
$794$	19.2983	0.684870
$795$	0	0
$796$	8.25945	0.292748
$797$	43.9094	1.55535	0.777675	−	0.628666i	$-0.216399\pi$
$797$	43.9094	1.55535	0.777675	+	0.628666i	$0.216399\pi$
$798$	8.30330	0.293934
$799$	33.6655	1.19100
$800$	0	0
$801$	1.51388	0.0534902
$802$	16.7250	0.590579
$803$	43.9441	1.55075
$804$	−4.86248	−0.171487
$805$	0	0
$806$	−25.2005	−0.887651
$807$	0.994977	0.0350248
$808$	5.39194	0.189688
$809$	−53.3794	−1.87672	−0.938360	−	0.345659i	$-0.887655\pi$
$809$	−53.3794	−1.87672	−0.938360	+	0.345659i	$0.887655\pi$
$810$	0	0
$811$	−7.53806	−0.264697	−0.132348	−	0.991203i	$-0.542252\pi$
$811$	−7.53806	−0.264697	−0.132348	+	0.991203i	$0.542252\pi$
$812$	2.52444	0.0885904
$813$	9.07451	0.318257
$814$	−21.5577	−0.755598
$815$	0	0
$816$	12.5436	0.439114
$817$	−9.15667	−0.320351
$818$	−26.4842	−0.925997
$819$	2.26499	0.0791451
$820$	0	0
$821$	4.08217	0.142469	0.0712343	−	0.997460i	$-0.477306\pi$
$821$	4.08217	0.142469	0.0712343	+	0.997460i	$0.477306\pi$
$822$	21.3970	0.746305
$823$	34.2338	1.19332	0.596658	−	0.802496i	$-0.296495\pi$
$823$	34.2338	1.19332	0.596658	+	0.802496i	$0.296495\pi$
$824$	−17.6116	−0.613530
$825$	0	0
$826$	6.58388	0.229082
$827$	−45.9935	−1.59935	−0.799676	−	0.600432i	$-0.794995\pi$
$827$	−45.9935	−1.59935	−0.799676	+	0.600432i	$0.794995\pi$
$828$	−1.64280	−0.0570912
$829$	−6.30330	−0.218923	−0.109461	−	0.993991i	$-0.534913\pi$
$829$	−6.30330	−0.218923	−0.109461	+	0.993991i	$0.534913\pi$
$830$	0	0
$831$	1.14663	0.0397761
$832$	3.10278	0.107569
$833$	−4.33804	−0.150304
$834$	22.3225	0.772964
$835$	0	0
$836$	−5.28917	−0.182930
$837$	39.9305	1.38020
$838$	−6.94610	−0.239949
$839$	40.6902	1.40478	0.702391	−	0.711791i	$-0.252116\pi$
$839$	40.6902	1.40478	0.702391	+	0.711791i	$0.252116\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−2.71585	−0.0935945
$843$	56.1588	1.93421
$844$	−12.2302	−0.420982
$845$	0	0
$846$	1.40753	0.0483919
$847$	−6.29776	−0.216394
$848$	5.30833	0.182289
$849$	−11.4947	−0.394498
$850$	0	0
$851$	−41.9945	−1.43955
$852$	10.3033	0.352985
$853$	52.1250	1.78473	0.892363	−	0.451319i	$-0.149046\pi$
$853$	52.1250	1.78473	0.892363	+	0.451319i	$0.149046\pi$
$854$	−25.4161	−0.869722
$855$	0	0
$856$	4.30833	0.147256
$857$	−48.5608	−1.65880	−0.829402	−	0.558652i	$-0.811319\pi$
$857$	−48.5608	−1.65880	−0.829402	+	0.558652i	$0.811319\pi$
$858$	−16.4111	−0.560266
$859$	−43.6308	−1.48866	−0.744332	−	0.667810i	$-0.767232\pi$
$859$	−43.6308	−1.48866	−0.744332	+	0.667810i	$0.767232\pi$
$860$	0	0
$861$	38.4211	1.30939
$862$	16.2978	0.555104
$863$	10.5839	0.360279	0.180140	−	0.983641i	$-0.442345\pi$
$863$	10.5839	0.360279	0.180140	+	0.983641i	$0.442345\pi$
$864$	−4.91638	−0.167259
$865$	0	0
$866$	−10.1169	−0.343787
$867$	55.9250	1.89931
$868$	−20.5033	−0.695928
$869$	5.05995	0.171647
$870$	0	0
$871$	−8.31889	−0.281875
$872$	−7.57331	−0.256465
$873$	0.367248	0.0124294
$874$	−10.3033	−0.348514
$875$	0	0
$876$	−27.3275	−0.923310
$877$	45.6938	1.54297	0.771485	−	0.636248i	$-0.219514\pi$
$877$	45.6938	1.54297	0.771485	+	0.636248i	$0.219514\pi$
$878$	9.79445	0.330547
$879$	−51.5421	−1.73847
$880$	0	0
$881$	4.55721	0.153536	0.0767682	−	0.997049i	$-0.475540\pi$
$881$	4.55721	0.153536	0.0767682	+	0.997049i	$0.475540\pi$
$882$	−0.181370	−0.00610705
$883$	−45.6222	−1.53531	−0.767654	−	0.640864i	$-0.778576\pi$
$883$	−45.6222	−1.53531	−0.767654	+	0.640864i	$0.778576\pi$
$884$	21.4600	0.721777
$885$	0	0
$886$	−21.7980	−0.732319
$887$	24.8122	0.833111	0.416555	−	0.909110i	$-0.363237\pi$
$887$	24.8122	0.833111	0.416555	+	0.909110i	$0.363237\pi$
$888$	13.4061	0.449878
$889$	5.35668	0.179657
$890$	0	0
$891$	28.5335	0.955910
$892$	11.2544	0.376826
$893$	8.82774	0.295409
$894$	3.62721	0.121312
$895$	0	0
$896$	2.52444	0.0843356
$897$	−31.9688	−1.06741
$898$	−22.2686	−0.743111
$899$	−8.12193	−0.270882
$900$	0	0
$901$	36.7144	1.22313
$902$	−24.4741	−0.814899
$903$	−23.1155	−0.769234
$904$	−5.60806	−0.186521
$905$	0	0
$906$	−16.1361	−0.536085
$907$	−14.8277	−0.492347	−0.246174	−	0.969226i	$-0.579173\pi$
$907$	−14.8277	−0.492347	−0.246174	+	0.969226i	$0.579173\pi$
$908$	−19.7647	−0.655916
$909$	1.55918	0.0517148
$910$	0	0
$911$	3.40753	0.112896	0.0564482	−	0.998406i	$-0.482022\pi$
$911$	3.40753	0.112896	0.0564482	+	0.998406i	$0.482022\pi$
$912$	3.28917	0.108915
$913$	38.2686	1.26650
$914$	−29.8958	−0.988864
$915$	0	0
$916$	−23.7250	−0.783895
$917$	9.44279	0.311828
$918$	−34.0036	−1.12229
$919$	2.59392	0.0855656	0.0427828	−	0.999084i	$-0.486378\pi$
$919$	2.59392	0.0855656	0.0427828	+	0.999084i	$0.486378\pi$
$920$	0	0
$921$	41.1320	1.35534
$922$	33.2927	1.09644
$923$	17.6272	0.580207
$924$	−13.3522	−0.439254
$925$	0	0
$926$	13.1184	0.431096
$927$	−5.09273	−0.167267
$928$	1.00000	0.0328266
$929$	8.56420	0.280982	0.140491	−	0.990082i	$-0.455132\pi$
$929$	8.56420	0.280982	0.140491	+	0.990082i	$0.455132\pi$
$930$	0	0
$931$	−1.13752	−0.0372806
$932$	24.0625	0.788193
$933$	−16.4806	−0.539550
$934$	−4.15165	−0.135846
$935$	0	0
$936$	0.897225	0.0293267
$937$	−0.498289	−0.0162784	−0.00813920	−	0.999967i	$-0.502591\pi$
$937$	−0.498289	−0.0162784	−0.00813920	+	0.999967i	$0.502591\pi$
$938$	−6.76830	−0.220993
$939$	45.4544	1.48335
$940$	0	0
$941$	13.1028	0.427138	0.213569	−	0.976928i	$-0.431491\pi$
$941$	13.1028	0.427138	0.213569	+	0.976928i	$0.431491\pi$
$942$	−31.7194	−1.03347
$943$	−47.6756	−1.55253
$944$	2.60806	0.0848850
$945$	0	0
$946$	14.7244	0.478733
$947$	5.67107	0.184285	0.0921424	−	0.995746i	$-0.470628\pi$
$947$	5.67107	0.184285	0.0921424	+	0.995746i	$0.470628\pi$
$948$	−3.14663	−0.102198
$949$	−46.7527	−1.51766
$950$	0	0
$951$	−27.1128	−0.879193
$952$	17.4600	0.565881
$953$	−31.9824	−1.03601	−0.518007	−	0.855377i	$-0.673326\pi$
$953$	−31.9824	−1.03601	−0.518007	+	0.855377i	$0.673326\pi$
$954$	1.53500	0.0496975
$955$	0	0
$956$	−4.31335	−0.139504
$957$	−5.28917	−0.170975
$958$	42.7144	1.38004
$959$	29.7834	0.961755
$960$	0	0
$961$	34.9658	1.12793
$962$	22.9355	0.739471
$963$	1.24583	0.0401464
$964$	−27.7144	−0.892621
$965$	0	0
$966$	−26.0100	−0.836860
$967$	−14.0484	−0.451765	−0.225882	−	0.974155i	$-0.572527\pi$
$967$	−14.0484	−0.451765	−0.225882	+	0.974155i	$0.572527\pi$
$968$	−2.49472	−0.0801833
$969$	22.7491	0.730808
$970$	0	0
$971$	46.7436	1.50007	0.750037	−	0.661396i	$-0.230036\pi$
$971$	46.7436	1.50007	0.750037	+	0.661396i	$0.230036\pi$
$972$	−2.99498	−0.0960639
$973$	31.0716	0.996110
$974$	18.3728	0.588702
$975$	0	0
$976$	−10.0680	−0.322270
$977$	32.8016	1.04942	0.524708	−	0.851282i	$-0.324175\pi$
$977$	32.8016	1.04942	0.524708	+	0.851282i	$0.324175\pi$
$978$	−5.49472	−0.175702
$979$	−15.2680	−0.487969
$980$	0	0
$981$	−2.18996	−0.0699202
$982$	−0.881639	−0.0281342
$983$	−47.5225	−1.51573	−0.757866	−	0.652411i	$-0.773758\pi$
$983$	−47.5225	−1.51573	−0.757866	+	0.652411i	$0.773758\pi$
$984$	15.2197	0.485186
$985$	0	0
$986$	6.91638	0.220262
$987$	22.2851	0.709342
$988$	5.62721	0.179025
$989$	28.6832	0.912074
$990$	0	0
$991$	3.00052	0.0953145	0.0476573	−	0.998864i	$-0.484824\pi$
$991$	3.00052	0.0953145	0.0476573	+	0.998864i	$0.484824\pi$
$992$	−8.12193	−0.257872
$993$	−32.8752	−1.04326
$994$	14.3416	0.454888
$995$	0	0
$996$	−23.7980	−0.754069
$997$	−42.3502	−1.34124	−0.670622	−	0.741799i	$-0.733973\pi$
$997$	−42.3502	−1.34124	−0.670622	+	0.741799i	$0.733973\pi$
$998$	32.4791	1.02811
$999$	−36.3416	−1.14980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.a.s.1.3	yes	3
5.2	odd	4		1450.2.b.k.349.4		6
5.3	odd	4		1450.2.b.k.349.3		6
5.4	even	2		1450.2.a.q.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.q.1.1	✓	3	5.4	even	2
1450.2.a.s.1.3	yes	3	1.1	even	1	trivial
1450.2.b.k.349.3		6	5.3	odd	4
1450.2.b.k.349.4		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.81361 q^{3} +1.00000 q^{4} +1.81361 q^{6} +2.52444 q^{7} +1.00000 q^{8} +0.289169 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.81361 q^{3} +1.00000 q^{4} +1.81361 q^{6} +2.52444 q^{7} +1.00000 q^{8} +0.289169 q^{9} -2.91638 q^{11} +1.81361 q^{12} +3.10278 q^{13} +2.52444 q^{14} +1.00000 q^{16} +6.91638 q^{17} +0.289169 q^{18} +1.81361 q^{19} +4.57834 q^{21} -2.91638 q^{22} -5.68111 q^{23} +1.81361 q^{24} +3.10278 q^{26} -4.91638 q^{27} +2.52444 q^{28} +1.00000 q^{29} -8.12193 q^{31} +1.00000 q^{32} -5.28917 q^{33} +6.91638 q^{34} +0.289169 q^{36} +7.39194 q^{37} +1.81361 q^{38} +5.62721 q^{39} +8.39194 q^{41} +4.57834 q^{42} -5.04888 q^{43} -2.91638 q^{44} -5.68111 q^{46} +4.86751 q^{47} +1.81361 q^{48} -0.627213 q^{49} +12.5436 q^{51} +3.10278 q^{52} +5.30833 q^{53} -4.91638 q^{54} +2.52444 q^{56} +3.28917 q^{57} +1.00000 q^{58} +2.60806 q^{59} -10.0680 q^{61} -8.12193 q^{62} +0.729988 q^{63} +1.00000 q^{64} -5.28917 q^{66} -2.68111 q^{67} +6.91638 q^{68} -10.3033 q^{69} +5.68111 q^{71} +0.289169 q^{72} -15.0680 q^{73} +7.39194 q^{74} +1.81361 q^{76} -7.36222 q^{77} +5.62721 q^{78} -1.73501 q^{79} -9.78389 q^{81} +8.39194 q^{82} -13.1219 q^{83} +4.57834 q^{84} -5.04888 q^{86} +1.81361 q^{87} -2.91638 q^{88} +5.23527 q^{89} +7.83276 q^{91} -5.68111 q^{92} -14.7300 q^{93} +4.86751 q^{94} +1.81361 q^{96} +1.27001 q^{97} -0.627213 q^{98} -0.843326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - q^{19} + 12 q^{21} + 5 q^{22} - 8 q^{23} - q^{24} + 2 q^{26} - q^{27} + 2 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{32} - 15 q^{33} + 7 q^{34} + 14 q^{37} - q^{38} + 4 q^{39} + 17 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} - 8 q^{46} + 12 q^{47} - q^{48} + 11 q^{49} + 11 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 2 q^{56} + 9 q^{57} + 3 q^{58} + 16 q^{59} + 2 q^{61} + 4 q^{62} - 18 q^{63} + 3 q^{64} - 15 q^{66} + q^{67} + 7 q^{68} + 6 q^{69} + 8 q^{71} - 13 q^{73} + 14 q^{74} - q^{76} - 4 q^{77} + 4 q^{78} - 13 q^{81} + 17 q^{82} - 11 q^{83} + 12 q^{84} - 4 q^{86} - q^{87} + 5 q^{88} + 11 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{93} + 12 q^{94} - q^{96} + 24 q^{97} + 11 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 + 2 * q^7 + 3 * q^8 + 5 * q^11 - q^12 + 2 * q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - q^19 + 12 * q^21 + 5 * q^22 - 8 * q^23 - q^24 + 2 * q^26 - q^27 + 2 * q^28 + 3 * q^29 + 4 * q^31 + 3 * q^32 - 15 * q^33 + 7 * q^34 + 14 * q^37 - q^38 + 4 * q^39 + 17 * q^41 + 12 * q^42 - 4 * q^43 + 5 * q^44 - 8 * q^46 + 12 * q^47 - q^48 + 11 * q^49 + 11 * q^51 + 2 * q^52 - 6 * q^53 - q^54 + 2 * q^56 + 9 * q^57 + 3 * q^58 + 16 * q^59 + 2 * q^61 + 4 * q^62 - 18 * q^63 + 3 * q^64 - 15 * q^66 + q^67 + 7 * q^68 + 6 * q^69 + 8 * q^71 - 13 * q^73 + 14 * q^74 - q^76 - 4 * q^77 + 4 * q^78 - 13 * q^81 + 17 * q^82 - 11 * q^83 + 12 * q^84 - 4 * q^86 - q^87 + 5 * q^88 + 11 * q^89 - 4 * q^91 - 8 * q^92 - 24 * q^93 + 12 * q^94 - q^96 + 24 * q^97 + 11 * q^98 - 6 * q^99

Introduction

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